The next few months, even weeks, will be telling. If major league players intend to oust Tony Clark as the head of their union, they probably will want to act sooner rather than later.

Players are upset by the difficulties many free agents experienced in the open market. An unknown percentage of them—and a number of agents—are agitating to remove Clark, believing the union needs to be led by an attorney and not a former player, sources say.

None of the five players or five agents who expressed discontent with Clark in recent interviews with The Athletic was willing to go on record with their criticisms of the union’s executive director, who became the first former player to assume that position following the death of Michael Weiner in Nov. 2013.

Players and agents generally like and respect Clark, and do not want to embarrass him. Many believe the union would benefit from Clark’s intelligence, passion and playing background in another role even if he...